Homework 1, due September 27.
To receive full credit, you must give the correct answer and show that you understand it. This requires writing your explanations in full, complete English sentences, clearly labeling all figures and graphs, showing us how you did arithmetic, and being explicit about the units of all numbers given. All relevant mathematical symbols should be explicitly defined. And please use your best handwriting; if I can't read it, I can't give you credit for it!
In this class you'll see some problems to which a good answer is basically verbal, some for which a fairly precise calculation is needed, and some for which ``order-of-magnitude" accuracy is all that's appropriate. It's usually up to you to figure out which is called for. As you will see, astronomers and mathematicians have rather different approaches to numbers. In particular, it is wrong to simply write down all the digits that your calculator spits out; if the numbers going into a given calculation are imprecise, the result will have a similar imprecision. Indeed, do the problems without a calculator whenever possible.
Feel free to work with your classmates on this homework, but your write-up should be your own. Answer all questions.
100 points total (plus 10 points extra credit)
1. How many stars? (50 points)
In this problem, we will get a sense of both how big, and how empty, the universe is.
(a) (10 points). The Milky Way galaxy in which we live is shaped like an enormous circular flattened disk. It has a radius of 40,000 light years, and a thickness of about 1000 light years. The mean distance between stars is about 4 light years. Calculate how many stars there are in the Galaxy. Hint: Think about how much volume the typical star, plus the empty space around it, occupies.
(b) (10 points). The Milky Way is only one of many similar galaxies in the universe. The mean distance between luminous galaxies is about light years. The visible universe has a radius of about 14 billion light years. Calculate how many galaxies there are in the visible universe.
(c) (5 points). Using the results of parts (a) and (b), calculate the total number of stars in the visible universe. Express your answer both in exponential notation, and for fun, in English (check out http://mathworld.wolfram.com/LargeNumber.html if you are unfamiliar with the names of really large numbers).
(d) (10 points). Let's go back to that number of four light years between stars. That is a seriously big number. Spacecraft in the solar system travel at roughly the same speed that the Earth travels around the Sun. First, calculate the latter speed (the Earth goes around the Sun, a circle of radius 1 Astronomical Unit = 150 million kilometers, in one year), and express your answer in kilometers per second. Second, knowing that light travels at 300,000 kilometers per second, calculate how many kilometers there are in four light years. Finally, calculate how long it would take for a spacecraft to reach the nearest stars. Express your answer in years.
(e) (10 points). Space is really empty as well. Calculate the ratio of the diameter of stars to the distance between them. Now imagine scaling the whole problem down: imagine the Sun as a basketball. At that scale, how far away is the nearest star? Express your answer in kilometers.
(f) (5 points). Almost done. Let us now calculate the density of matter in the Milky Way. The sun has a mass of grams. Calculate the total mass of the Milky Way (assuming, of course, that all stars have the same mass), and then calculate its density (i.e., the total mass divided by the total volume; see part a), in units of grams per cubic centimeter. Compare with the density of the most extreme laboratory vacuum () and comment.
2. Expanding Universe, Expanding Oceans (10 points)
The expansion of the universe discovered by Hubble really only takes place on physical scales larger than galaxies, but for the purposes of this exercise, imagine that the proportionality between distance and recession velocity holds on scales comparable to the size of the Earth. Calculate by what distance the Atlantic Ocean (width 7000 km) would grow in a year under the expansion of the universe. Compare with the actual growth, due to plate tectonics (a completely different physical mechanism), which adds up to several inches a year. Which is larger?
3. Trying to see planets (40 points, plus 10 points extra credit)
Here we will start to see why taking pictures of planets around other stars is a difficult matter...
(a) (10 points). Consider a planet orbiting 5 Astronomical Units from its parent star, which is 10 light years from Earth. Using the small angle formula, calculate the angular distance on the sky between the planet and the star. Express your answer in arcseconds. (For simplicity, assume that the orbit of the planet is in the plane of the sky).
(b) (10 points). The planet itself, like Jupiter, has a radius of 70,000 kilometers. Calculate its angular size in arcseconds. The best telescopes on the ground can resolve (i.e., measure the angular size of) objects no less than 0.3 arcseconds in angular size. Compare this number to the two numbers you've just calculated and comment.
(c) (10 points). The star, like the Sun, has a surface temperature of 6000 K, and a radius of 700,000 kilometers. The planet has a surface temperature of 200 K. Calculate the ratio of the luminosity of the star and the planet, assuming each radiate like black bodies, and ignoring the effects of reflected light. Hint: do as much algebra as possible before plugging in numbers!
(d) (10 points). You will find above that the star is enormously more luminous than is the planet; it's like looking for a firefly next to a searchlight. But the situation isn't quite that bad: calculate the wavelengths at which the spectra of the star and the planet each peak, and comment. Express your answer in microns (one micron = meters).
(e) (10 points). For extra credit, recalculate the ratio of the luminosity of the star and planet, now per unit wavelength at the peak wavelength of the planet, and comment.Michael Strauss